In order to construct a hydrodynamic theory we need a way to relate averages of currents to that of charges: we need an equation of state. In statistical mechanics, this information is fully encoded within the “free energy flux", first considered in [1]}. This is a function of the state \(\mathcal {G}\)  whose derivatives give rise to average currents
\(\langle \hat{j}_n\rangle = \frac{\partial \mathcal {G}}{\partial \beta _n}\;.\) 
